We develop a combinatorial condition necessary for the existence of a saturated asymmetrical orthogonal array of strength 2. This condition limits the choice of integral solutions to the system of equations in the
Bose-Bush approach and can thus strengthen considerably the Bose-Bush approach as applied to a symmetrical part of such an array. As a consequence, several nonexistence results follow for saturated and nearly saturated orthogonal
arrays of strength 2. One of these leads to a partial settlement of an issue left open in a paper by Wu, Zhang and Wang. Nonexistence of a class of saturated asymmetrical orthogonal arrays of strength 4 is briefly
discussed.